The charge of particles in dispersion is of importance for their stability, rheological properties, coating behavior and other things. The zeta potential ζ is defined as potential at the surface of shear where the particle with a shell of electrostatically attracted counter ions moves through the bulk solution of a sample. The value of the zeta potential is not equal to the surface potential because of the bound ions. However, it is a relevant potential for describing the particle interaction in dispersion.
The electrophoretic mobility μg, defined as the equilibrium velocity ν the particle attains in an electric field E, can be related to the zeta potential according to the Henry equation by:
                              μ          e                =                              v            E                    =                                                    2                ⁢                ζ                            ∈                              f                ⁡                                  (                                      κ                    ⁢                                                                                  ⁢                    α                                    )                                                                    3              ⁢                              η                ⁡                                  (                  T                  )                                                                                        (        1        )            
where η(T) is the viscosity as a function of the absolute temperature T, ∈ is the dielectric constant of the dispersing medium, and ƒ(κα) is a function of the particle size α and the thickness of the double layer, the so-called Debye length 1/κ.
The Smoluchowski approximation is valid for moderate ion concentrations and not too small particle size where κα>100 and ƒ(κα) becomes 1.5.
For small particles in non-polar medium, where κα<1, the Hückel approximation is valid. In this case ƒ(κα) becomes 1.
For intermediate values of κα, ƒ(κα) can only be determined numerically.
Laser Doppler Electrophoresis LDE (or Electrophoretic Light Scattering (ELS)) is an established technique for measuring the electrophoretic mobility of dispersed particles which is based on light scattering. In a suited arrangement of light source, detector, and electric field the scattered light experiences a frequency shift caused by the well-known Doppler-effect. In order to make this shift, which is small compared to the absolute light frequency, measureable the scattered light is mixed with a reference beam. The interference of the two contributions results in a beat with the Doppler-shift frequency.
Often a phase modulator is applied to the reference beam in order to create an additional frequency shift. This allows distinguishing between positive and negative frequency shifts corresponding to a positive and a negative sign of the zeta potential. Moreover, shifting the origin of the frequency spectrum to a non-zero value improves the stability and accuracy of the measurement of small, near-zero mobilities. The setup becomes less prone to thermal and mechanical changes.
The omnipresent random diffusive motion of the dispersed particles superimposes the collective electrophoretic motion. While the collective motion causes a Doppler shift, the diffusion causes a broadening of the spectral peak which in turn limits the accuracy of the measured Doppler frequency. Since the width of the spectral peak increases with q2, whereas the Doppler shift scales only with q, it is beneficial to measure LDE at small scattering angles (small values of q). Here q is the magnitude of the scattering vector defined as:
                    q        -                                            4              ⁢              π              ⁢                                                          ⁢              n                        λ                    ⁢                      sin            ⁡                          (                              θ                2                            )                                                          (        2        )            
with n being the refractive index of the dispersing agent, λ the wavelength of the incident beam, and θ the scattering angle.
At small scattering angles, however, the Doppler shift becomes small and long measurement times are needed in order to achieve sufficiently good statistical accuracy.
The cell walls carry charge, thus the application of an electric field causes the liquid adjacent to the wall to undergo electro-osmotic flow. However, in a closed system the flow along the walls must be compensated for by a reverse flow in the center of the cell. Dispersed particles will be subject to this flow superimposed on their electrophoretic mobility. Thus, in order to measure the electrophoretic velocity alone electro-osmotic effects have to be avoided. In addition electrode polarization, electrolysis and Joule heating may cause errors as well and have to be avoided too.
There are several approaches which proved to be effective.
First, there is a certain position in the sample cell where the electro-osmotic flow at the cell wall and the reverse flow in the center of the cell cancel. At this position, which is called the stationary layer, the particle velocity is unbiased by electroosmosis.
Second, the sign of the applied electric field can be reversed fast enough to avoid the formation of an electroosmotic flow, while the electrophoretic motion still reaches its equilibrium velocity. This means that the measured mobility is due to electrophoresis only and is not affected by electroosmosis. Fast reversal of the electric field also minimizes effects of electrode polarization and electrolysis. On the other hand, however, this effectively breaks the temporal averaging procedure of the signal into many short time batches.
In order to keep the energy input into the sample (the Joule heating) as small as possible the magnitude of the electric field has to be kept as low as possible. This in turn means that small Doppler shifts have to be measured.
Phase Analysis Light Scattering (PALS) is a modification of Laser Doppler Electrophoresis which makes it possible determining small frequency shifts by measuring a series of short intervals.
Rather than analyzing the beat frequency (for instance by Fourier transformation), PALS is looking at the change of phase with time. Obviously, this rate is equivalent with the frequency. However, recasting the problem in this way makes it possible to evaluate signals where only a fraction of a Doppler cycle is available and greatly increases the sensitivity for small velocities of the particles.
PALS makes it possible to make use of the short measurement intervals needed to avoid electroosmosis, electrode polarization, and electrolysis by fast reversal of the electric field. The statistical accuracy is reached by averaging many such short intervals. Nevertheless even when PALS is used, it is still beneficial to keep the time intervals which are measured in a phase-locked way as long as possible.
In PALS the rate of phase change of the measured interference between scattered beam from the sample and the modulated reference beam is analyzed. This rate is compared with a mathematically generated sine wave predetermined by the modulator frequency.
Any non-linearity of the modulator and any change in the characteristics of the modulator (for instance because of a change in temperature, change in the frequency or aging) will cause a situation where the mathematically generated frequency does not reflect the real conditions any more. Ideally, the generated wave corresponds to the beat frequency when the particles in the sample are not moving. The electrophoretic mobility and in turn the zeta potential is determined from the frequency difference (difference in rate of phase change) due to the Doppler shift. Thus, any error in the mathematically generated frequency translates into an error of the zeta potential.
For Laser Doppler Electrophoresis and also for PALS it is beneficial to produce the interference between scattered beam from the sample and the modulated reference beam for a long period of time in a phase-locked way. Thus, the use of a modulator with a large phase-range is increasing the stability and accuracy of the measurement.
Modulators with a large total phase-range are usually not linear. This means for instance for a piezo or voice-coil driven modulator that a linearly increasing voltage or current does not cause a linear motion of the modulator. Such a non-linearity of the modulator is resulting in a temporal change of the beat frequency during the move. In addition the characteristics of such a modulator also vary with temperature, frequency and age. This behavior is not compatible with the standard PALS method because no simple function can be generated which is correctly describing the characteristics of the modulator.
Conventional systems of determination of the electrophoretic mobility are disclosed in US 2011/0210002 A1, WO 2010/041082 A2, U.S. Pat. No. 7,295,311 B2, J. F. Miller, K. Schätzel, and B. Vincent, “The determination of very small electrophoretic mobilities in polar and nonpolar colloidal dispersions using phase analysis light scattering”, Journal of Colloid and Interface Science, 1991, 143(2): p. 532-554, and F. McNeil-Watson, W. Tscharnuter, and J. Miller, “A new instrument for the measurement of very small electrophoretic mobilities using phase analysis light scattering (PALS)”, Colloids and Surfaces A: Physicochemical and Engineering Aspects, 1998, 140(1-3): p. 53-57.
In conventional apparatuses, a mathematically generated wave is used for the demodulation of the Laser Doppler Electrophoresis signal. Modulator non-linearity causes an error in the determination of the electrophoretic mobility. Temporal changes of the modulator characteristics, for instance with temperature, frequency or age, cause an error in the measured electrophoretic mobility.